A home team backed by a capacity crowd at the Mohammed V Stadium had virtually given up hope of cracking open a brave Nigerian defence when they were awarded an 89th-minute free-kick.
Enyimba goalkeeper Paul Godwin, who had been outstanding under a relentless Wydad onslaught led by Congo-Brazzaville striker Fabrice Ondama, misjudged the kick and diminutive 25-year-old Benin midfielder Angan nodded home.
Godwin, who committed a rare blunder to hand Heartland victory in the Nigerian FA Cup final last weekend, was lucky not to be sent off a minute after the goal when he handled outside the box.
It was the first loss suffered by two-time African champions Enyimba in the competition this year after eight victories and three draws, including one at the same Casablanca stadium against Raja during the group phase.
Enyimba host the return match on 16 October in the south-eastern city Aba with 1992 champions Wydad seeking an away goal.
Esperance of Tunisia, runners-up to TP Mazembe of Congo DR last year, are favoured to overcome Al-Hilal of Sudan in the other semi-final with Omdurman hosting the first leg Sunday.